------------- While we are asking terminological questions prompted by Mac Lane and Moerdijk, I'd like to know exactly what people mean by the Yoneda lemma. Does Yoneda include the fact that every presheaf is a canonical limit of representables? Does it include (as Johnstone has it, on _Topos Theory_ p.51) the claim that the Yoneda embedding to a presheaf category is left adjoint to the forgetful functor (i.e. the functor that forgets the action of arrows and only remembers the family of sets indexed over the objects of the domain category)? Colin McLarty ==============================================================================
In the phrase "Yoneda lemma" the first word is generic. The connection with Yoneda is as follows. I wrote a letter to David Buchsbaum in 1959 in which I used one of the lemmas now known as Yoneda's. (I had seen the lemma in an abstract in the Notices by Watts.) Barry Mitchell, in a return letter, mentioned that the lemma was Yoneda's. In the book Abelian Categories I give a reference to a paper by Yoneda for the lemma. Some years after the publication of the book a student complained to me that the lemma does not appear in the cited paper. The complaint was duly passed on to Barry. He refered to his notes from the lectures that Mac Lane had given in the summer of 58 or 59 (I think) on Yoneda's treatment of the Ext functors, in which notes the lemma does appear. Barry had assumed that the lemma was in the paper that Saunders was reporting on. And, of course, I never thought of actually looking at a paper I was citing. But Yoneda must have known the Yoneda lemma. One may describe the Yoneda lemma as case zero of his theorems on Ext. The lemma certainly needs a name and Yoneda sounds nice. In that original formulation it is just the lemma that the maps from a functor represented by an object, A, to an arbitrary set-valued functor, F, are in natural corespondence with the elements of F(A). (Well, not quite: the original formulation was for additive categories and instead of "set-valued" it was "group-valued".) Almost immediately the phrase was generalized. I remember John Gray in the early 60's talking about the importance of the Yoneda lemma in enriched category settings. I don't know what Yoneda has to say about all of this. If he had stayed in pure mathematics he probably would have proved too many things to make the name useful as the sole description of a single lemma. As it is, though, the name serves well. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Peter's comments on the Yoneda lemma are interesting. Phil Scott was in Japan some years ago and actually spoke to Yoneda and it would be interesting to hear what he has to say. Computer scientists are fascinated by the fact that I actually know Kleisli and that he is still around. And that he actually did do the Kleisli construction. Michael ==============================================================================
participants (3)
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barr@triples.Math.McGill.CA -
cxm7@pop.cwru.edu -
pjf@saul.cis.upenn.edu